Identifiability up to a Declared Invariance Group: Hidden-State Models Outside Phylogenetics

Question: How have identifiability and congruence problems been addressed in hidden-state and state-dependent Markov models outside phylogenetics — such as hidden Markov models with unobserved states, or mixture models with label switching?

Answer

Across the retrieved literature, the field has largely abandoned the phylogenetic aspiration of proving that parameters are uniquely recoverable, and has replaced it with three narrower moves. First, identifiability is now stated relative to a named invariance group — state permutation, scaling, sign similarity, eigenspace rotation — and the decision problem itself has been formalised and complexity-classified rather than assumed away [3] [2]. Second, where genuine identifiability is claimed, it is purchased from auxiliary structure: multi-lag temporal dependence, regime nonstationarity, exponential-family or non-Gaussian noise, structural sparsity, or marginal independence, typically yielding recovery only up to permutation and scaling [12] [5] [21]. Third, label switching is treated as a problem about the estimand — the posterior functional being summarised — rather than a defect in the model, via factorised posteriors, optimal-transport discrepancies between mixing measures, and modular decoupling of classification from downstream inference [7] [8] [10]. Taken together, these papers support a deflationary position: congruence between hidden states recovered under different fits, subjects, or model families is not delivered by identifiability theory but is engineered post hoc, and the field has no shared finite-sample measure of it.

Identifiability is now defined relative to an explicit invariance group, so congruence becomes a declared property rather than a hidden defect

The HMM identifiability question has been reformulated computationally: the decision problems arising from deterministic, generic, global, local, state-permutation-invariant, and finite-alphabet notions are all decidable in PSPACE via reductions to the theory of the reals, and the deterministic variants are already coETR-hard for simply parameterised families [3]. The same paper states that most existing identifiability conditions are sufficient but not necessary, so a model failing them remains inconclusive rather than provably unidentifiable [3]. In a different latent-variable family, the non-identifiability gap is characterised exactly by three invariances — scale, sign similarity, and eigenspace rotation — with a dimension-counting theorem establishing additional continuous non-identifiability whenever a binomial coefficient falls below N(N+1)/2 [2]. For probabilistic transition systems, robust analogues replace exact equivalence: universal perturbation equivalence admits a polynomial-time partition-refinement algorithm and is NL-complete to decide, whereas existential language equivalence for distributions is complete for the existential theory of the reals [14]. Identifiability guarantees can also be tied to a distributional choice: linear recovery of latents holds for stationary additive-noise transition worlds, and the Gaussian is shown to be the unique latent distribution for which that guarantee holds [18].

Temporal dependence and nonstationarity are the dominant substitutes for external labels, and they deliver identifiability only up to permutation and scaling

For multi-lag regime-switching models, identifiability of both the number of regimes and the multi-lag transitions is proved in a nonlinear-Gaussian setting, while for switching dynamical systems latent variables are identifiable up to permutation and scaling through temporal structure, which in turn yields conditions for regime-dependent latent causal graphs up to regime and node permutations [12]. Markov switching models with instantaneous effects, nonlinear lagged effects, and independent exponential-family noise are shown to be identifiable in latent regimes and regime-dependent causal structure under temporal regime dependence, and the result subsumes non-temporal mixtures of causal models [5]. Semi-parametric state-space models that replace logistic transition links with learned functions in an RKHS or spline space establish identifiability conditions plus a consistency argument for the resulting estimators [15]. A non-temporal route exists as well: unlabelled finite mixtures whose components are marginally independent on at least one coordinate pair are identifiable under subset-rank, full-rank, and no-cancellation conditions, with the mixing matrix recoverable under stated completion conditions [21].

Where structure is insufficient, the field redefines the estimand rather than repairing the model

Amortised Bayesian inference for mixtures factorises the posterior into a parameter distribution and a categorical mixture-indicator distribution, explicitly motivated by label switching, and thereby sidesteps the labelling problem by learning membership classifiers instead of interpreting component indices [7]. A decision-theoretic alternative takes the mixing measure itself as the inference target and defines the loss as a sliced-Wasserstein discrepancy between mixing measures, an approach advertised as model-agnostic and valid under arbitrary dependence in the sampling model [8]. Modular Bayesian inference decouples classification from downstream inference using fixed class probabilities, with calibration conditions under which the procedure preserves the inferential target of an analysis with known labels [10]. Non-identifiability can even be exploited: parameter-expanded data augmentation deliberately constructs a non-identifiable model so that a restricted covariance matrix need not be sampled, improving MCMC convergence and mixing relative to the identifiable formulation [17]. For finite mixtures with nonparametric components, conditions are given under which individual component distributions are identified, with posterior contraction for component densities at a nearly polynomial rate, contrasted against logarithmic rates for mixing measures via deconvolution [9].

Detectability, state-count selection, and parameter recovery come apart, and the gaps are quantified rather than closed

In binomial logistic mixtures, observed-data evidence for two components and per-observation information for label recovery have different local orders in component separation, producing a detectable-but-unrecoverable regime where BIC selects two components while posterior labels remain essentially uninformative; recoverability-aware BIC with a posterior-entropy penalty and an entropy-regularised estimator are proposed in response [11]. State-count selection is similarly unstable in the nonparametric setting: a hidden hierarchical Dirichlet process construction recovers the true latent structure in simulation whereas the classical infinite HMM tends to overestimate the number of states when transitions vary over time [1]. Weak identifiability is reported in practice for specific emission parameters — the Singh–Maddala scale parameter — even under a fully specified Bayesian HMM with simulation-verified recovery of other parameters [16]. Inference about weakly identified quantities is nonetheless possible in a distributional sense: with unidentifiable nuisance parameters under the null, boundary parameters, and singular information, the likelihood ratio statistic converges to the supremum of a chi-bar-squared process with weights depending on the nuisance parameter [20].

Congruence across subjects and fits is enforced by hierarchical sharing and post hoc alignment, not by identifiability

A joint nonparametric Bayesian extension of the hierarchical Dirichlet process autoregressive HMM with subject-specific transition parameters is reported to give consistent hidden state estimation across multiple subjects and repeated measurements without pre-specifying the number of states [6]. In heterogeneous system identification, cluster identities are inferred in one shot by measuring alignment between leading covariance eigenspaces, with finite-sample bounds on the probability of pairwise false eigenspace alignment and sufficient conditions for uniform inter-cluster separation [4]. Where congruence itself is the scientific quantity, alignment is computed post hoc: the correspondence between children's dynamic brain states and averaged adult states is measured and used as a predictor of reading comprehension performance [19].

Conflicts and gaps

The papers disagree about how much identifiability buys. One line proves identifiability of the number of regimes in nonlinear-Gaussian multi-lag models [12], while another documents routine overestimation of the state count by infinite HMMs under time-varying transitions [1] — generic identifiability and practical state selection are evidently different problems. The claimed equivalence classes are also not comparable: permutation plus scaling [12], permutation up to regime and node relabelling [5], component-wise identifiability only under structural sparsity [13], and up-to-linear-map recovery under Gaussian latents [18] are derived under incompatible assumptions, and no retrieved paper ranks their strength. The decision procedure of [3] is shown decidable in PSPACE but exponential-time in effect, and no paper connects its verdicts to finite-sample recovery, so the practical value of a verdict remains untested. The label-switching remedies [7] [8] [10] change the estimand and are not identifiability results; none reports whether labels so obtained are congruent across independent fits or datasets. Finally, misspecification is handled only in approximation: approximate subspace identifiability is derived when precise conditions fail [13], and irreducibility is noted to be, in general, not directly testable from unlabelled mixtures alone [21].

What would change the answer

A congruence benchmark would settle most of the above: simulate from known HMMs and Markov switching models, refit independently across replicates, subsamples, and subjects, and score label agreement against truth after Hungarian matching — comparing entropy-penalised selection [11], sliced-Wasserstein posterior summaries [8], amortised membership classifiers [7], and hierarchical sharing [6] on the same data. Running the PSPACE decision procedure of [3] on standard HMM benchmarks and correlating its verdicts with realised recovery error would test whether inconclusive sufficient conditions matter empirically. Extending the perturbation-equivalence partition refinement of [14] from labelled Markov chains to HMMs with emissions would give robust equivalence classes under estimated rather than known transition probabilities. Testing the Gaussian-uniqueness prediction of [18] inside regime-switching latent dynamics — injecting non-Gaussian latent noise and checking whether linear identifiability breaks — would directly probe the assumption on which that guarantee rests. A cross-family test that fits an identifiable Markov switching model [5] and a switching dynamical system [12] to identical data and asks whether the recovered regime partitions agree would be the first direct measurement of congruence between two identifiability theories that currently make non-overlapping claims.

References

[1] Giuseppe Mignemi, Daniele Tancini, Francesco Bartolucci. "Generalized Infinite Hidden Markov Models". arxiv. 2026-10-23. URL: https://arxiv.org/abs/2610.31337

[2] Hideitsu Hino, Keisuke Yano. "From DPPs to $k$-DPPs: identifiability analysis via spectral decomposition". arxiv. 2026-05-26. URL: https://arxiv.org/abs/2605.25526

[3] Markel Zubia, Nils Jansen. "On the Computational Complexity of Hidden Markov Model Identification". arxiv. 2026-10-08. URL: https://arxiv.org/abs/2610.09104

[4] Abdulmoneam Ali, Dipankar Maity, Ahmed Arafa. "Eigenspace-Based Clustering for Personalized System Identification". arxiv. 2026-10-01. URL: https://arxiv.org/abs/2606.20811

[5] Roel Hulsman, Carles Balsells-Rodas, Sara Magliacane. "Identifiable Markov Switching Models with Instantaneous Effects and Exponential Families". arxiv. 2026-06-15. URL: https://arxiv.org/abs/2606.02231

[6] Fei W, Miao Z, Xu T, Wang Y. "Joint Bayesian Hidden Markov Model With Subject-Specific Transitions for Wearable Sensor Data.". pubmed. 2025-12-01. URL: https://doi.org/10.1002/sim.70334

[7] Šimon Kucharský, Paul Christian Bürkner. "Amortized Bayesian Mixture Models". arxiv. 2026-03-17. URL: https://arxiv.org/abs/2501.10229

[8] Khai Nguyen, Peter Mueller. "Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures". arxiv. 2025-05-09. URL: https://arxiv.org/abs/2411.14674

[9] Yilei Zhang, Yun Wei, Aritra Guha, XuanLong Nguyen. "A Bayesian approach to learning mixtures of nonparametric components". arxiv. 2026-03-06. URL: https://arxiv.org/abs/2512.12988

[10] Maximilian Autenrieth, Konstantin Karchev, David A. van Dyk, Kaisey Mandel, Roberto Trotta. "Modular Bayesian Inference for Partially Misspecified Mixtures with Uncertain Class Labels". arxiv. 2026-10-21. URL: https://arxiv.org/abs/2610.27085

[11] Yuta Hayashida, Shonosuke Sugasawa. "Information Gap and Feasibility-Aware Inference in Binomial Logistic Mixtures". arxiv. 2026-06-16. URL: https://arxiv.org/abs/2606.15665

[12] Carles Balsells-Rodas, Toshiko Matsui, Pedro A. M. Mediano, Yixin Wang, Yingzhen Li. "On the Identifiability of Regime-Switching Models with Multi-Lag Dependencies". arxiv. 2026-01-08. URL: https://arxiv.org/abs/2601.03325

[13] Yuke Li, Yujia Zheng, Ziyi Chen, Kun Zhang, Heng Huang. "Towards Identifiable Representations under Misspecified Structure". arxiv. 2026-09-29. URL: https://arxiv.org/abs/2609.33273

[14] Syyeda Zainab Fatmi, Stefan Kiefer, James C. A. Main, David Parker. "Perturbation equivalence in labelled Markov chains". arxiv. 2026-09-15. URL: https://arxiv.org/abs/2609.13401

[15] Prakul Sunil Hiremath. "Learning Nonlinear Regime Transitions via Semi-Parametric State-Space Models". arxiv. 2026-04-08. URL: https://arxiv.org/abs/2604.04963

[16] Safaa K. Kadhem. "A Bayesian Hidden Markov Model with Singh--Maddala Emission Distributions for Heavy-Tailed Regime-Switching Time Series". arxiv. 2026-10-19. URL: https://arxiv.org/abs/2610.22656

[17] Zhang X. "Parameter-expanded data augmentation for analyzing multinomial probit models.". pubmed. 2025-01-01. URL: https://doi.org/10.1080/03610926.2025.2559116

[18] David Klindt, Yann LeCun, Randall Balestriero. "When Does LeJEPA Learn a World Model?". arxiv. 2026-05-27. URL: https://arxiv.org/abs/2605.26379

[19] Zhang, J., Liu, L., Chen, J., Zhao, N., Li, H., Yang, X., Meng, X., Ding, G.. "Brain State Dynamics and Developmental Differences in Reading Comprehension". biorxiv. 2026-08-20. URL: https://doi.org/10.64898/2026.08.12.744344

[20] Karl Oskar Ekvall, Ola Hössjer, Matteo Bottai, J. M. Patrik Albin. "Asymptotics for likelihood ratio tests of boundary points with singular information and unidentifiable nuisance parameters". arxiv. 2026-10-15. URL: https://arxiv.org/abs/2605.08471

[21] Takafumi Kanamori, Yushi Hirose, Shohei Yamamoto. "Identifiability and Estimation for Unlabeled Finite Mixtures under Marginal Independence". arxiv. 2026-08-14. URL: https://arxiv.org/abs/2606.07914

How this survey was made

This survey answers one question: “How have identifiability and congruence problems been addressed in hidden-state and state-dependent Markov models outside phylogenetics, such as hidden Markov models with unobserved states or mixture models with label switching?”

We keep our own collection of recent preprints and refresh it every day from bioRxiv, medRxiv, arXiv, PsyArXiv, SocArXiv and ChemRxiv. For each survey, we search that collection for recent work and keep the strongest matches: papers that use the same terms as the question, and papers on the same topic in different words.

The text was written from the titles and abstracts of those papers; the full texts were not read. Statements that draw on a particular paper are marked with a number in brackets. The reference list just above is the 21 papers whose work this text draws on, chosen from the 136 papers those searches returned; the rest of the shortlist is not printed. It is not a complete review of everything published on the question.

Treat this as a starting point and a reading list, not a peer-reviewed review. The searches behind this survey ran on October 29, 2026 and covered work from the previous two years.